General Algebra (Summary of Algebra 1 & 2) Study Guide

General Algebra (Summary of Algebra 1 & 2) Study Guide Topic 1: Fundamentals of Algebra ● Basics of Solving Equations: ○ Vocabulary ■ Numerical Expression: an expression is a phrase used in math that has numbers and operations, but no an equal sign ■ Equation: a mathematical sentence that uses an equal sign; separates two equivalent expressions ■ Solution: a value that takes place of a variable value, making an equation true ■ Operations: Addition, Subtraction, Multiplication, Division, etc. ■ Properties of Equality: properties proving that if you do an operation on both sides of an equation, the statement will remain true ○ How to Solve Two-Step Equations 1. Isolate the variable (move everything else to the other side of the equal sign) 2. Use the opposite of PEMDAS to solve (SADMEP) - think of this like unraveling the equation a. Everything you do to one side you must also do to the other 3. Simplify all the way until you have a numerical solution to your variable ■ Example: 6x - 4 = 12 6x - 4 + 4 = 12 + 4 6x = 16 6x/6 = 16/6 x = 8/3 ● Set of Numbers: ○ Vocabulary ■ Roster Notation: all numbers in a set listed between braces, {} ■ Interval Notation: () are used to exclude the endpoint of an interval while [] are used to include the endpoint of an interval ex. (2, 6) all real numbers between but not including 2 and 6 [1, 3] all real numbers between and including 1 and 3 ● Square Roots: ○ Vocabulary ■ Radical Symbol: a symbol that shows that the number on the right squared is equal to the number inside the radical √36 = 6 ○ Properties of Square Roots ■ Product Property of Square Roots: √𝑎𝑏 = √𝑎 ⋅ √𝑏 𝑎 𝑎 ■ Quotient Property of Square Roots: √𝑏 = √𝑏 √ ● Simplifying Algebraic Expressions: ○ Content ■ To evaluate expressions with given values of the variables, substitute the numbers into the correct variable and use PEMDAS to solve ■ To simplify algebraic expressions, first identify like terms by the variable and the exponents, then combine like terms. ● Exponents: ○ Vocabulary ■ Scientific Notation: a way of writing large numbers by using powers of 10 ex. 3.274 х 10⁶ = 3,274,000 ○ Properties of Square Roots ■ Zero Exponent Property: a⁰ = 1 ■ Negative Exponent Property: 𝑎 −𝑛 1 = (𝑎)𝑛 = 𝑎 −𝑛 ( ) 𝑏 ■ Product of Powers Property: am・an = am +n ■ Quotient of Powers Property: am/an = am-n ■ Power of a Power Property: (am)n = am・n ■ Power of a Product Property: (ab)m = ambm 1 𝑎 𝑛 𝑏 𝑛 = ( ) 𝑎 ■ Power of a Quotient Property: (a/b)m = am/bm ● Transformations: ○ Vocabulary ■ Transformation: changing a geometric figure by altering its size, shape or position in order to create a new figure ■ Translation: a type of transformation where a figure slides across a graph, keeping the size and shape of the figure ■ Reflection: a type of transformation where a figure is reflected over a line of reflection, typically over the x or y-axis ■ Stretch: pulling points away from the x or y-axis ■ Compression: pushing points towards the x or y-axis ○ Translations ■ Horizontal Translation: each point moves right or left (x, y) → (x + h, y) ■ Vertical Translation: each point moves up or down (x, y) → (x, y + k) ○ Reflections ■ Reflection across x-axis: y-coordinate changes (x, y) → (x, -y) ■ Reflection across y-axis: x-coordinate changes (x, y) → (-x, y) ○ Stretches and Compressions ■ Horizontal Stretch: ■ Vertical Stretch: (x, y) → (bx, y) (x, y) → (x, ay) ■ Horizontal Compression: (x, y) → (bx, y), ■ Vertical Compression: (x, y) → (x, ay) |b|>1 | a | >1 0< |b|<1 0<|a|<1 Topic 2: Basic and Linear Functions, Equations, Inequalities, and Systems ● Recognizing Functions: ○ Vocabulary ■ Function: a type of mathematical relationship where there can only be one output value for each input value ■ Domain: x values ■ Range: y values ○ Tips to Recognize Functions ■ The easiest way to figure out if something is a function or not is to collect the domain and see if there are any repeats in the group. If there are none, then it is a function since no two inputs are the same value! It is okay to have two of the same output ■ The Vertical Line Test: To check this on a graph, draw vertical lines through the function depicted. If a line crosses through the same point twice vertically, that means there are two of the same input, making it not a function! ○ Graphing Functions 1. Enter your X values into the table function of your calculator 2. Enter your Y values into the table as well 3. Specify what kind of function you are trying to graph and choose that (linear, exponential, quadratic, etc.) 4. You should have a graph of your function and be able to manipulate it on your screen ■ Depending on what calculator you have, the process may be different but these are general steps ■ After creating your graph, you can use it to find values and data points ● Parent Functions: ○ Vocabulary ■ Parent Function: the simplest functions that provide a definition for certain types of functions ○ Different Parent Functions ● Piecewise Functions: ○ Vocabulary ■ Piecewise Function: function with one or more functions combined ○ Transformations of Piecewise Functions ■ When transforming Piecewise Functions, use the same rules as transforming any functions ● Operations with Functions: ○ Vocabulary ■ Composition of Functions: a function that uses the result from one function as the input for another ○ Function Operations ■ Addition: (f + g)(x) = f(x) + g(x) ■ Subtraction: (f - g)(x) = f(x) - g(x) ■ Multiplication: (fg)(x) = f(x) • g(x) ■ Division: (f/g)(x) = f(x)/g(x) where g(x) ≠ 0 ■ Composition of Functions: (fоg)(x) = f(g(x)) ● Inverses of Relations and Functions: ○ Vocabulary ■ Inverse Relation: reflecting and graphing each point across the line y=x ■ Inverse Functions: Functions that reverse each other ○ Content ■ You can use the horizontal line test to see if the inverse relation is a function (similar to the vertical line test) ■ You can write the inverse function f (x) as f-1 (x) when the relation is a function ■ Use the inverse operation to get the inverse function. ● Linear Functions: ○ Vocabulary ■ Linear Equation: any equation that can be full expressed in this form ● Standard Form of a Linear Equation: Ax + By = C ■ Discrete Function: a function whose points are not all continuously connected when graphed or entered into a table ■ Continuous Function: a function whose points are all continuously connected when graphed or entered into a table ■ Y-Intercept: where a coordinate intersects the y-axis ■ X-Intercept: where a coordinate intersects the x-axis ■ Rate of Change: the rate at which the function’s points change, relationship where the change in y is divided by the change in x ○ Things to Know about Solving Linear Equations ■ Solutions are typically an ordered pair that can be graphed on a coordinate plane ■ They can also be in the form x = n or y = n (n being > 0) ● Determining Slope: ○ Vocabulary ■ Rise: difference between two y-values on a graphed line or in a table of points ■ Run: difference between two x-values on a graphed line or in a table of points ■ y = mx + b: the notorious slope equation, the ratio of the difference in yvalues to the difference in x-values ■ Point-Slope Form: used to find the equation of a line when given the slope and one of the coordinates (looks like this: y - y₁ = m(x - x₁) ) ● Linear Systems ○ Vocabulary ■ System of Linear Equations: two or more linear questions that share variables, the solution to a system is an ordered pair that satisfies both or all of the equations in the system ○ Different Kinds of Systems Consistent Inconsistent Independent Dependent System System System System A system that has A system that A system that has A system that has at least one doesn’t have a only one solution infinite solutions solution solution ○ Methods to Solving Linear Systems ■ elimination ● Graphing 1. Make sure all of the equations of the system are in the right format to be graphed 2. Either graph by hand or using a graphing calculator 3. Define your solutions by looking at which points are crossed by all involved equations ● Substitution 1. Solve one of the equations to find one of the variable’s values (is usually the equation inverted some way) 2. Plug that value into the other equation where the variable is 3. Solve the equation for the variable that dominates the new equation 4. After finding that variable’s numerical value, plug it into one of the other equations to find the other variables numerical value ● Elimination 1. Align like terms in the equations, prepare to cancel out one of the variable values 2. Multiply one of the equations if needed 3. Add/subtract them to cancel out one of the variables 4. Solve the equation for the leftover variable 5. Plug in the newfound numerical value for one of the variables into one of the original equations to find the other variable’s numerical value ● Inequalities: ○ Vocabulary ■ Variable: an unknown value in an expression, equation, or inequality, usually represented with a symbol or letter (ex. x) ■ Coefficient: a numerical value tied to an unknown value (a variable), if there is no number representing the coefficient, it is assumed to be 1 ■ Constant: a numerical value that remains fixed or constant, is not directly affected by the variables in the equation ■ Algebraic Expression: a mathematical phrase that contains values, variables and operations but no equal sign ○ Tips for Solving Inequalities ■ Solving inequalities is the same process as solving equations but there are some things to watch out for ■ When dividing or multiplying both sides of an inequality by a negative number, you must flip the sign (ex. < turns into >) ○ Deciphering Algebraic Expressions ■ In algebra, there are many expressions that are written multiple ways but all mean the same thing, here are a few examples (n=any number) ● n-6 = n-(-6) ● n/10 = n * 1/10 (or 0.1) ○ Inequality Symbols Symbol Verbal Description > Greater than < Less than ≥ Greater than or equal to ≤ Less than or equal to ≠ Not equal ○ Graphing Inequalities ○ Graphing Systems of Linear Inequalities ■ Graph the lines just like you would for linear equations. Then shade one side of the line, depending on your inequality’s sign (if the equation is greater than y - shade above, if it is less than - shade below) ■ Open Circle: Dashed/Dotted Line ■ Closed Circle: Regular Line ■ After shading all of the inequalities of the system, the area that is shaded by all of them is the solution ● Curve Fitting with Linear Models: ○ Vocabulary ■ Regression: statistical study of the relationship between variables to find trends ■ Correlation: a measure of the strength of the relationship between a pair of variables as well as what direction the relationship leans toward ■ Line of Best Fit: a line that best represents the data according to each point ■ Correlation Coefficient: represented by r-measures how well two variables fit the correlation (-1 ≤ r ≥ 1) ○ Correlation Coefficient ■ r = 1, straight line, positive slope ■ r = 0, no correlation ■ r = -1, straight line, negative slope ● Absolute-Value Equations and Inequalities: ○ Vocabulary ■ Disjunction: compound statement that joins two statements with the word OR (∪) ■ Conjunction: compound statement that joins two statements with the word AND (∩) ■ Absolute-value function: a function that contains an absolute value sign (parent absolute value function has a v-shape with its point at the origin) ● Linear Programming: ○ Vocabulary ■ Linear Programming: a way of finding the biggest or smallest value of a function that matches a given set of conditions called constraints ■ Constraint: one inequality in a problem; limits ■ Feasible Region: a graphed solution of a constraint ■ Objective Function: finds the best combination in a constraint whether it is a minimum or maximum, both or neither values ○ Content ■ Find the maximum value by identifying the vertices and putting each into the equation. Topic 3: Exponential Functions and Relationships ● The Basics of Radicals ○ Vocabulary ■ Radical Expression: and expression that has a radical, one that uses the square root operation (√) ■ Radicand: the number inside of the radical sign, usually referred to as n ■ Index: the number sitting on top of the tail of the racial sign, usually referred to as a ■ Rational Exponent: the number inside of the radical sign, usually referred to as n ○ Radical Properties ■ Product Property of Radicals: √ab = √a * √b ■ Quotient Property of Radicals: √a/b = √a/√b ● Exponential Functions, Growth, and Decay ○ Vocabulary ■ Asymptote: a line of a graphed function that could never be approached as the line continues ■ Exponential Function: a function where the outputs values are related to each other by a constant ratio; modeled with the parent function: y=abx ■ Exponential Growth: a function where a is greater than zero and b is greater than one; the values grow exponentially and rapidly ■ Exponential Decay: when b is a number between 0 and 1; the y- values decrease exponentially as the x-values get larger ○ Exponential Regression ■ A tool usually found on graphing calculators that helps you find the exponential function and whether it is growth or decay ■ Takes points and graphs them to form a line and equation for the regression Topic 4: Polynomial Functions and Operations ● Polynomials ○ Vocabulary ■ Polynomial Equations: an equation consisting of many terms with multiple different powers using the same variables ● Tri(nominal): consisting of three terms ● Poly(nominal): consisting of four or more terms ● Adding and Subtracting Polynomials ○ Remember! ■ Rewrite the equation, pairing like terms together and turning double subtraction into addition ● Multiplying Polynomials ○ Tips and Tricks ■ Making tables helps when multiplying a lot of terms ■ Utilize the Distributive Property ○ FOIL Method ■ Multiply the First terms ■ Multiply the Outer terms ■ Multiply the Inner terms ■ Multiply the Last terms ● Dividing Polynomials: ○ Vocabulary ■ Synthetic division: a way of dividing polynomials by linear binomial using coefficients instead of using long division ○ Content ● Factoring Polynomials: ○ Factor Theorem ■ You can use this theorem to determine whether a linear binomial is a factors ■ Use synthetic substitution to solve. The binomial is a factor of P(x) only if P(x) equals 0 ○ Factoring by Grouping ■ Another method to solve polynomials 1) Group terms 2) Factor out common variables or numbers from each group 3) Factor out common binomial 4) Factor the difference of squares ○ Factoring the Sum and the Difference of Two Cubes ■ Sum of two cubes: a3 + b3 = (a + b)(a2 - ab + b2) ■ Difference of two cubes: a3 - b3 = (a - b)(a2 + ab + b2) ● Finding Real Roots of Polynomial Equations ○ Vocabulary ■ Multiplicity: the number of times a root is repeated on a single point ○ Multiplicity ■ When a root has a multiplicity that is bigger than 1, the graph bends at the point of the root ■ After you have factored a polynomial equation, you can determine what multiplicity of a root has by seeing how many of the binomial appears ex. (x-3)(x-3)(x+4)➝root 3 has a multiplicity of 2 and -4 has a multiplicity of 1 ○ Rational Root Theorem ■ Every Rational Root of a Polynomial could be written in the form p/q, where p represents the factor of the polynomial and q is the factor of the leading coefficient of the polynomial ○ Irrational Root Theorem ■ When a + b√c is a root of a polynomial equation, then a - b√c is also a root of the polynomial P(x) = 0. ● Fundamental Theorem of Algebra: ○ The Fundamental Theorem of Algebra ■ Every polynomial with a degree of 1 or larger has at least one zero, where a zero could be a complex number ○ Complex Conjugate Root Theorem ■ If a + bi is a root of a polynomial equation, so is a - bi. ● Graphs of Polynomial Functions: ○ Vocabulary ■ End Behavior: the behavior of a polynomial function as x approaches positive or negative infinity ■ Turning Point: where a graph turns from decreasing to increasing or increasing to decreasing ■ Local Maximum: the highest point given a range ■ Local Minimum: the lowest point given a range ● Transforming Polynomial Functions: ○ Content Topic 5: Quadratic Functions and Inequalities ● Graphing Quadratic Functions ○ Vocabulary ■ Quadratic Function: a function that can be represented in this form: f(x) = ax2 + bx + c ■ Vertex Form: f(x) = a(x - h)2 + k , where (h, k) is the vertex ■ Quadratic Formula: ■ Parabola: the curve of a graphed quadratic function ■ Vertex: the point on the parabola where the graph changes direction ■ Axis of Symmetry: a line that divides the parabola in half, it passes through the vertex ● Found using the following equation: x = - b/2a ■ Minimum: the value of the vertex when it is the lowest point of the graph (when a > 0) ■ Maximum: the value of the vertex when it is the highest point of the graph (when a < 0) ○ Maximums and Minimums ○ Solving for Zeros ■ Zero: an x value that accounts for a parabola’s x-intercept value (x, 0) ■ Solve by factoring from the ax2 + bx + c to two factors ■ Utilize the Zero Product Property 1. Set each factor’s f(x) equal to zero 2. Solve each of the equations for x to find the zeros. ● Solving Quadratic Inequalities: ○ Vocabulary ■ Quadratic Inequality in two variables: inequalities that require two variables; they can be written in any of these forms: y≤ ax2 + bx + c yax2 + bx + c ○ Graphing Quadratic Inequalities 1) Graph the Quadratic inequality just like you would in a quadratic equation 2) Use solid line for y≤ and y≥ and a dashed line for y< and y> 3) Shade above the line for y≥ and y> and below for y≤ and y< ● Operations with Complex Numbers ○ Vocabulary ■ Complex Plane: a graph where the x-axis represents real numbers and the y-axis represents imaginary numbers ■ Absolute Value of a Complex Number: a + bi equals the distance from the point (a,b) in the complex plane Topic 6: Logarithmic Functions ● Logarithmic Functions: ○ Vocabulary ■ Logarithm: the quantity representing the power to which the base has to be raised in order to meet a certain number bx = a → logba = x ■ Common Logarithm: if there is no base of a logarithm, assume it’s 10 ■ Logarithmic Function: inverse of an exponential function ○ Content ■ Logarithm is basically the opposite of exponents: 26 = 64 → log264 = 6 ■ When solving for log, ask yourself: what exponent of the base will meet the power ■ Special Properties of Logarithms: Logarithmic Form Exponential Form Example Logarithm of Base b logbb = 1 b =b log2020 = 1 201 = 20 Logarithm of 1 logb1 = 0 b0 = 1 log101 = 0 100 = 1 1 ■ Exponential vs. Logarithmic Function ● Properties of Logarithms: ○ Product Property of Logarithms ■ The sum of the logarithms of its factors is equal to the logarithm of a product ■ log45000 = log4(50 ・100) = log450 + log4100 ■ logbmn = logbm + logbn ○ Quotient Property of Logarithms ■ The logarithm of a quotient= the difference between the logarithm of the dividend and the logarithm of the divisor 12 ■ log4( 2 ) = log4(12) - log4(2) 𝑚 ■ logb( 𝑛 ) = logbm - logbn ○ Power Property of Logarithms ■ The logarithm of a power equals the logarithm of the base ■ log104 log (10)(10)(10)(10) log10 + log10 + log10 + log10 4log10 ○ Inverse Properties of Logarithms and Exponents ■ logbbx = x ex. log10108 = 8 ■ blogbx = x ex. 10log⏨3 = 3 ○ Change of Base Formula ■ logbx = logax/logab ex. log5100 = log10100/log105 ● The Natural Base, e: ○ Vocabulary ■ Natural Logarithm: a logarithm with a base e (abbreviated “In”) ■ Natural Logarithmic Function: Inverse of the natural exponential function f(x) = ex natural log function→ (f(x) = In x) ○ Content ■ Use properties of logarithms and apply them to solve natural logarithms ● Transformation of Logarithmic Functions ○ Content Transformation Vertical Translation f(x) Notation f(x) + k Examples y = log x + 4 y = log x - 3 4 units up 3 units down Horizontal Translation Vertical Stretch Or Compression Horizontal Stretch or Compression f (x-h) y = log(x - 7) right y = log(x + 2) 7 units 2 units left af(x) y = 2 log x stretch by 2 y = ¼ log x compression by ¼ f(1/b x) y = log (⅙x) stretch by 6 y = log (2x) compression by ½ Topic 7: Probability, Sequences, and Statistics ● Patterns and Sequences: ○ Vocabulary ■ Sequence: a group of numbers that are listed in a specific order, due to some pattern that comes with each term ■ Term: a number in a sequence ■ Position Number: each number in a sequence has a position number based on what number of the sequence they are (ex. 7, 9, 11, 13...is a sequence, 11 is the 3rd number so 11’s position number is 3) ■ Explicit Rule: a rule that can be used to find the nth term of a sequence that does not have depending factors on previous terms, you do not need to find all of the previous terms to find some nth term (ex. f(n) = n2 + 2) ■ Recursive Rule: a rule used to find the nth term of a sequence that has factors depending on previous terms, you cannot skip around and must find every previous term before the nth term you are looking for, these are more complex than explicit rules (ex. f(n) = f(n - 1) + 3) ● Statistical Models: ○ Vocabulary ■ Quantitative Data: data that can be expressed using numerical values ■ Categorical Data: data that cannot be expressed using numerical values ■ Frequency: how often something appears in a set of data, is usually organized in a table ● Two-Way Frequency: frequencies with two variables ○ Relative Frequency ■ You can convert data to its relative frequency to know what part or percentage belongs to a certain group ■ Relate frequency can be found by dividing the amount in the group by the total amount of frequencies Joint Relative Frequency Marginal Relative Conditional Relative Frequency Frequency A percentage that tells you A percentage that tells you A percentage that tells you what portion of the total data what portion of the data fits what portion of the one fits into both of the two into one of the categories that category’s data fits into categories that are being are being looked at another category looked at ● Measures of Center ○ Vocabulary ■ Mean: the sum of the values in a data set divided by the number of values in the set ■ Range: the difference between the biggest and smallest value points ■ Median: the value within the middle of a data set ○ Box Plots ■ Whisker: lines extending from the box of a box plot, connecting it to the minimum and maximum values ■ Minimum: the smallest value of data displayed on a box plot and in a set of data ■ Lower/First Quartile: the median between the minimum and the actual median ■ Interquartile Range (IQR): the difference between the third and first quartile data points ■ Upper/Third Quartile: the median between the actual median and the maximum ■ Maximum: the largest value of data displayed on a box plot and in a set of data ■ Outlier: a data point that is exceedingly far from the other data points, usually leads to the skewing of data ● Standard Deviation ○ Vocabulary ■ Mean Standard Deviation: the square root of the average distance away from the mean by each data point ● Normal Distribution ○ Vocabulary ■ Normal Distribution: data that when displayed, is symmetric, looks like a bell-shaped curve with tails on either side, a normal curve ● Scatter Plots and Trend Lines ○ Vocabulary ■ Scatter Plot: a visual showcasing two-variable data ■ Two-Variable Data: a collection of paired values such as ordered pairs ■ Line of Best Fit: a line that is drawn through a scatter plot, showing the correlation of the two-variable data ■ Residual: a vertical measure of the space between a data point and the line of best fit ■ Linear Regression: the most widely used method to find the least-squares line ■ Least-Squares Line: line that represents the sum of the squared residuals as the smallest it can be ● Permutations and Combinations: ○ Vocabulary ■ Fundamental Counting Principle: a method to figure out the number of outcomes of an event; you label the number of ways to choose an item as m1・m2・.... mn when n represents the item ■ Permutation: listing all the ways a group of items could be placed ■ Factorial: Multiplying every natural number under a number until one including itself (5! = 5・4・3・2・1 = 120) ■ Combination: grouping items in no certain order ○ Content ● Theoretical and Experimental Probability: ○ Vocabulary ■ Probability: how likely something is to occur ■ Theoretical Probability: ratio of the number of favorable outcomes to the total number of outcomes for equally likely outcomes ■ Complement: the probability of event A’s outcomes except itself (P(not A) = 1 - P(A) ■ Geometric Probability: probability solved by a ratio of lengths, areas, or volumes ■ Experimental Probability: the quotient of the number of times the event occurs and the number of trials ● Independent and Dependent Events: ○ Vocabulary ■ Independent Events: an event by itself that does not affect another event’s probability ■ Dependent Events: when the occurence of one event affects the probability of another event ■ Conditional Probability: states how to solve the measure of the probability of an event given that another event happened ○ Probability of Independent Events ■ When A and B are both independent events, then P(A and B) = P(A)・P(B) ○ Probability of Dependent Events ■ When both A and B are dependent events, P(A and B) = P(A)・P(B | A), where P(B | A) is the probability of B, given that A has occurred ● Compound Events: ○ Vocabulary ■ Simple Event: an event with one outcome ■ Compound Event: several simple events ■ Mutually Exclusive Events: events that cannot happen at the same time in an experiment ■ Inclusive Events: events that can happen simultaneously ● Mutually Exclusive Events ○ Inclusive Events ● Binomial Distributions: ○ Vocabulary ■ Binomial Theorem: an easy way to expand any binomial using Pascal’s triangle ■ Binomial Experiment: experiment that has independent trials with successful or failing outcomes ■ Binomial Probability: P(r) = nCrprqn-r ● Series and Summation Notation: ○ Vocabulary ■ Partial Sum: sum of terms in part of the sequence ■ Summation Notation: a sign that helps with effectively adding sequences ○ Content ● Arithmetic Sequences and Series: ○ Vocabulary ■ Arithmetic Sequence: numbers in which the difference between those terms are constant ■ Arithmetic Series: sum of terms in an arithmetic sequence ○ Content ● Mathematical Induction and Infinite Geometric Series: ○ Vocabulary ■ Infinite Geometric Series: series that have infinitely many terms ■ Converge: when a series is |r| < 1 and the partial sum approaches a fixed number ■ Limit: the number that the partial sums approach as n increases ■ Diverge: when a series is |r| ≥ 1 and the partial sum does not approach a fixed number ■ Mathematical Induction: a type of mathematical proof to prove formula you used for sums ○ Content ■ Sum of an Infinite Geometric Series: S = a1/1-r’ when r is the common ratio and |r| < 1 is where a1 is the first term Topic 8: Rational and Radical Functions ● Rational Functions: ○ Vocabulary ■ Rational Function: a function that could be written as a ratio of two polynomials within which its graph is a hyperbola ○ Content ) ● Radical Expressions and Rational Exponents: ○ Vocabulary ■ Index: the n of a radical expression n√a ■ Rational Exponent: an exponent that is a fraction a1/n = n√a ● Radical Functions: ○ Vocabulary ■ Radical Function: a function that includes a radical expression ■ Square-root Function: a radical function with a square root ○ Content ● Radical Equations: ○ Vocabulary ■ Radical Equation: an equation that contains a variable inside a radical ○ Solving Radical Equations 1) Isolate the radical/variable 2) Undo the radical by raising both sides of the equation to the power equal to the index of the radical 3) Simplify and solve Topic 9: Matrices ● Matrices and Data: ○ Vocabulary ■ Matrix: a rectangular array of numbers inside a bracket ■ Dimensions: “m by n”; m rows and n columns ■ Scalar: Multiplying a matrix by a number ○ Adding and Subtracting Matrices ■ You can only add or subtract matrices only if they have the same dimensions ■ Add or subtract the number in the matrix corresponding to the spot its in ■ ［2 3］+［4 1］= ［6 4］ ○ Scalar ■ You multiply every number in the matrix by the scalar to get the scalar product ● Multiplying Matrices: ○ Vocabulary ■ Matrix Product: multiplying two or more matrices ○ Multiplying Matrices ■ You can only multiply matrices when the number of columns of the first and the number of rows in the second are equal ■ ● Determinants and Cramer’s Rule: ○ Vocabulary ■ Determinant: difference of the products of the diagonals of a 2 by 2 matrix ○ Cramer’s Rule for Two Equations ■ Topic 10: Conic Sections ● Conic Sections: ○ Vocabulary ■ Conic Sections: a section formed where a double right cone and a plane intersect ○ Content ■ You can get the following shapes in a conic section: circles, ellipses, hyperbolas, and parabolas ● Circles: ○ Vocabulary ■ Tangent: a line that touches only a single point of a circle ○ Equation of a Circle ■ (x - h)2 + (y - k)2 = r2 ■ (h, k) is the center of the circle and r is the radius ● Hyperbolas: ○ Vocabulary ■ Hyperbola: a symmetrical curve formed by the intersection of two equal cones ○ Content ■ How the standard form of the hyperbola is written determines whether the hyperbola’s transverse axis is horizontal or vertical ■ Horizontal Equation: 𝑥2 𝑦2 − 2 2 𝑎 𝑏 = 1 ■ Vertical Equation: ● Identifying Conic Sections: ○ Content ■ 𝑦2 𝑥2 − 2 2 𝑎 𝑏 = 1 Topic 11: Trigonometric Functions, Graphs and Identities ● Right-Angle Trigonometry ○ Vocabulary ■ Trigonometric Function: a function that relates to a ratio of sides of a right triangle ■ Sine: opposite/hypotenuse ■ Cosine: adjacent/hypotenuse ■ Tangent: opposite/adjacent ■ Cosecant: 1/sinθ = hypotenuse/opposite ■ Secant: 1/cosθ = hypotenuse/adjacent ■ Cotangent: 1/tanθ = adjacent/opposite ● The Unit Circle: ○ Vocabulary ■ Radian: unit of angle measure of an arc when arc length is equal to the radius ■ Unit Circle: a circle that has a radius of 1 ○ Converting Angle Measures ■ Degrees to radians: multiply degrees by (π radians/180°) ■ Radians to Degrees: multiply radians by (180°/π radians ○ Using Reference Angles to Evaluate Trigonometric Functions 1) Find the reference angle θ (smallest positive angle made with the x-axis) 2) Get the sine, cosine, and tangent by applying the reference angle 3) Change the signs according to the original quadrant of the angle if needed ● The Law of Sines: ○ Area of a Triangle ■ You can use these formulas to solve for the area of a triangle when you don’t know the height of the triangle ■ For triangle ABC, where abc are sides opposite to the angles ABC: ■ Area = ½bc sinA ■ Area = ½ bc sin B ■ Area = ½ab sin C ○ Law of Sines ■ For triangle ABC, with sides abc, the Law of Sines says that: sinA = sinB = sinC a b c ● The Law of Cosines: ○ Content ■ For triangle ABC, where corresponding sides of the angles are abc, Law of Cosines states: ■ a2 = b2 + c2 - 2bccosA ■ b2 = a2 + c2 - 2accosB ■ c2 = a2 + b2 - 2abcosC ○ Heron’s Formula ● Graphs of Trigonometric Functions: ○ Vocabulary ■ Periodic Functions: functions that repeat cycles in regular intervals ■ Amplitude: half of the difference of the maximum and minimum values of sine and cosine functions ■ Frequency: a way to measure sound by the number of cycles in a given unit of time ● Fundamental Trigonometric Identities: ○ Reciprocal Identities ■ cscθ = 1/sinθ ■ secθ = 1/cosθ ■ cotθ = 1/tanθ ○ Tangent and Cotangent Ratio Identities ■ Tanθ = sinθ/cosθ ■ cotθ = cosθ/sinθ ○ Pythagorean Identities ■ cos2θ + sin2θ = 1 ■ 1 + tan2θ = sec2θ ■ cot2 + 1 = csc2θ ○ Negative-Angle Identities ■ sin(-θ) = -sinθ ■ cos(-θ) = cosθ ■ tan(-θ) = -tanθ ● Sum and Difference Identities: ○ Vocabulary ■ Rotation Matrix: rotating a point through an angle θ using matrices ○ Sum Identities ■ sin(A + B) = sinAcosB + cosAsinB ■ cos(A + B) = cosAcosB - sinAsinB ■ tan(A + B) = tanA +tanB/1 - tanAtanB ○ Difference Identities ■ sin(A - B) = sinAcosB - cosAsinB ■ cos(A - B) = cosAcosB + sinAsinB ■ tan(A - B) = tanA -tanB/1 + tanAtanB ○ Rotation Matrix